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Parameterization of 3D Conformal

Transformations in Conformal Geometric

Algebra

Hongbo Li

Abstract Conformal geometric algebra is a powerful mathematical language for

describing and manipulating geometric conﬁgurations and their conformal trans-

formations. By providing a 5D algebraic representation of 3D geometric conﬁg-

urations, conformal geometric algebra proves to be very helpful in pose estima-

tion, motion design, and neuron-based machine learning (Bayro-Corrochano et al.,

J. Math. Imaging Vis. 24(1):55–81, 2006; Dorst et al., Geometric Algebra for Com-

puter Science, Morgan Kaufmann, San Mateo, 2007; Hildenbrand, Comput. Graph.

29(5):795–803, 2005; Lasenby, Computer Algebra and Geometric Algebra with Ap-

plications, LNCS, vol. 3519, pp. 298–328, Springer, Berlin, 2005;Lietal.,Geo-

metric Computing with Clifford Algebras, pp. 27–60, Springer, Heidelberg, 2001;

Mourrain and Stolﬁ, Invariant Methods in Discrete and Computational Geometry,

pp. 107–139, Reidel, Dordrecht, 1995; Rosenhahn and Sommer, J. Math. Imaging

Vis. 22:27–70, 2005; Sommer et al., Computer Algebra and Geometric Algebra with

Applications, pp. 278–297, Springer, Berlin, 2005). In this chapter, we present some

theoretical results on conformal geometric algebra which should prove to be useful

in computer applications. The focus is on parameterizing 3D conformal transfor-

mations with either quaternionic Vahlen matrices or polynomial Cayley transform

from the Lie algebra to the Lie group of conformal transformations in space.

1 Terminology and Notations

By embedding Euclidean space R

into the set of null vectors in R

n+1,1

in a nonlin-

ear manner, we get the conformal model of nD Euclidean geometry [2, 5, 6, 11]. In

the Minkowski space R

n+1,1

, a nonzero vector is said to be null if its inner product

with itself is zero and is said to be positive if so is the inner product.

H. Li (



)

Mathematics Mechanization Key Laboratory, Academy of Mathematics and Systems Science,

Chinese Academy of Sciences, Beijing 100080, China

e-mail: hli@mmrc.iss.ac.cn

E. Bayro-Corrochano, G. Scheuermann (eds.), Geometric Algebra Computing,

DOI 10.1007/978-1-84996-108-0_4, © Springer-Verlag London Limited 2010

72 H. Li

A point in R

is represented by a null vector in R

n+1,1

, and the representation

is unique up to scale. There is a unique-up-to-scale null vector e ∈ R

n+1,1

in the

conformal model that does not represent any point in R

. We say that it represents

the conformal point at inﬁnity. A sphere or hyperplane in R

is represented by a

positive vector in R

n+1,1

, and the representation is unique up to scale. A positive

vector represents a hyperplane if and only if its inner product with e equals zero.

The Grassmann–Cayley algebra [19] over R

n+1,1

, when equipped with the nD

Euclidean geometric interpretations of the algebraic elements in this algebra, is

called the conformal Grassmann–Cayley algebra of the nD space. The Clifford al-

gebra over R

n+1,1

, when equipped with the nD conformal transformation interpre-

tations of the algebraic elements in this algebra, is called the conformal Clifford

algebra of the nD space. Conformal geometric algebra is an integration of con-

formal Grassmann–Cayley algebra and conformal Clifford algebra [9], the former

representing geometric conﬁgurations, and the latter representing geometric trans-

formations.

Terminology and notation:

1. Tensor product, denoted by “⊗”.

2. Outer product, denoted by “∧”.

3. Inner product, denoted by “·”.

4. Meet product, denoted by “∨”.

5. Geometric product, denoted by juxtaposition of participating elements. The ge-

ometric product of r identical elements A is denoted by A

6. Multivector: any element in a Grassmann algebra (or Clifford algebra).

7. Exponential of a multivector A:exp(A) =e

=1 +A +A

/2!+A

/3!+···.

8. Inverse of a multivector A, denoted by A

−1

9. Blade: a multivector which equals the outer product of several vectors.

10. Grade: the number of vector components in an outer product decomposition of

a blade. A blade of grade r is called an r-blade.

11. Homogeneous multivector: a linear combination of blades of the same grade.

The grade of a homogeneous multivector is that of any of the blade component.

A homogeneous multivector of grade r is called an r-vector. A 2-vector is also

called a bivector.

12. r-graded part of a multivector, denoted by “

”.

13. Scalar part of a multivector: the 0-graded part, denoted by “”.

14. Even (or odd) multivector: a linear combination of homogeneous multivectors

of even grades (or odd grades).

15. Even-graded part (or odd-graded part) of a multivector, denoted by “

”(or

“

−

”).

16. Vers or : the geometric product of several invertible vectors.

17. Rotor: the geometric product of an even number of invertible vectors.

18. Positive vector: a vector whose inner product with itself is positive.

19. Positive versor: the geometric product of several positive vectors.

20. Positive rotor: the geometric product of an even number of positive vectors.

21. Grassmann algebra over V

, denoted by Λ(V

22. Clifford algebra over V

, denoted by C(V

Parametrization of 3D Conformal Transformations 73

23. Grassmann algebra or Clifford algebra generated by a blade I

and denoted

by Λ(I

) (or C(I

)). The base vector space V

of the Grassmann algebra (or

Clifford algebra) is composed of all vectors whose outer product with I

equals

zero.

24. Even Clifford subalgebra of C(V

), composed of all even multivectors, de-

noted by C

25. Odd vector subspace of C(V

), composed of all odd multivectors, denoted by

C

−

26. Magnitude of a multivector: denoted by “||”. The magnitude of a scalar is its

absolute value. The magnitude of A ∈C(V

) is |A|=



i=0

√

|A

·A

27. Pseudoscalar: a blade of grade n in Λ(V

) or C(V

28. Dual operator in a nondegenerate Clifford algebra, denoted by “

∼

”. Fixing a

pseudoscalar I

of unit magnitude, for any multivector A, A

∼

=AI

−1

29. Reversion operator in a Clifford algebra, denoted by “

†

”. For vectors a

, a

,...,

, (a

···a

)

†

···a

30. Grade involution in a Clifford algebra: denoted by overhat. For multivector A,



A =A

−A

−

31. Conjugate operator in a Clifford algebra: the composition of reversion and

grade involution, denoted by overbar. For multivector A,

A =A

†



−A

†



−

32. Spin group over V

, denoted by Spin(V

). It is composed of all rotors in

C(V

) of unit magnitude together with the geometric product.

Example 1 In conformal Grassmann–Cayley algebra C(R

4,1

), a circle passing

through three points 1, 2, 3 in the space is represented by 3-blade 1 ∧2 ∧3, where

1, 2, 3 are null vectors of R

4,1

representing points in R

. The blade is Minkowski, so

its dual (1 ∧2 ∧3)

∼

is a positive vector. Alternatively, the circle can be represented

by a positive vector.

The set N of all null vectors in R

n+1,1

has two connected components. In partic-

ular, null vectors ±a are always in different connected components, as 0 is not a null

vector. An orthogonal transformation in R

n+1,1

keeping each component of N in-

variant is called a positive orthogonal transformation. All such transformations form

a subgroup O

(n +1, 1) of O(n +1, 1), called the positive orthogonal group.The

orientation-preserving orthogonal transformations of R

n+1,1

form another subgroup

SO(n +1, 1) of O(n+1, 1), called the special orthogonal group. The intersection

of the two subgroups, denoted by SO

(n +1, 1), is called the Lorentz group, and its

elements are called Lorentz transformations. Lorentz transformations are the linear

isometries of R

n+1,1

connected with the identity transformation I

n+1,1

In conformal Clifford algebra C(R

n+1,1

), any positive rotor U induces a unique

Lorentz transformation in R

n+1,1

via the following adjoint action:

(x) =UxU

−1

for all x ∈R

n+1,1

. (1)

Conversely, any Lorentz transformation in R

n+1,1

is induced by a positive rotor that

is unique up to scale.

74 H. Li

In the conformal model, any Lorentz transformation in R

n+1,1

induces a unique

orientation-preserving conformal transformation in R

, and the converse is also true.

Hence, any orientation-preserving conformal transformation in R

is induced by a

positive rotor in C(R

n+1,1

) that is unique up to scale.

2 Exponential Map and Exterior Exponential Map

By a classical theorem of Riesz [14], any linear isometry of R

n+1,1

connected with

the identity is induced by a rotor of the exponential form. The group of rotors in

C(R

n+1,1

) differs from Spin(R

n+1,1

) by a factor R −{0}. Since the Lie algebra of

the spin group is all bivectors in Λ(R

n+1,1

), any rotor connected with the identity

can be expressed up to scale as the exponential e

of a bivector B

∈Λ(V

).Asa

corollary, any positive rotor in C(R

n+1,1

) is in the range of the exponential map.

Example 2 The Lie algebra representation of 3D rigid body motions via the expo-

nential map.

Any rigid body motion in the space can be decomposed into a rotation followed

by a translation. It can also be decomposed into a translation followed by a rota-

tion. The decomposition is not unique without ﬁxing the axis of rotation, which is

a straight line in the space. However, there is a unique decomposition, in which the

axis of rotation follows exactly the direction of translation. This is the screw motion,

and the unique decomposition theorem is known as Chasles’ Theorem.

In the conformal model of 3D Euclidean geometry, let e

, e

be an orthonor-

mal basis of R

, and let e

, e be the pair of null vectors orthogonal to R

in R

4,1

and such that e

·e =−1. The basis (e, e

, e

) is called a Witt basis of R

4,1

The vector e

represents the origin of R

in the conformal model, while the vector

e represents the conformal point at inﬁnity.

In conformal geometric algebra C(R

4,1

), the Lie algebra of the spin group of

rigid body motions is Λ

∼

), which is the bivector subspace of the Grassmann

algebra generated by the vectors in R

4,1

that are orthogonal to e.

The vector space Λ

∼

) has an orthonormal basis e

, e

, ee

Any nonzero element in Λ

∼

) can be written as

θ +et

, (2)

where 2-blade I

∈Λ(R

) is of unit magnitude, θ ∈R, and t ∈ R

If θ =0, then e

induces the translation by the vector t.Ifθ =0, then

= e

−e(t·I

)/(2θ)

θ/2

e(t·I

)/(2θ)

⊥

(t)/2

= cos

sin

(t) sin

⊥

(t) cos

⊥

(t)I

sin

, (3)

Parametrization of 3D Conformal Transformations 75

where

(t) =(t ·I

−1

⊥

(t) =t −P

(t).

(4)

Equation (3) induces a screw motion with the vector of translation P

⊥

(t), the axis

of rotation passing through the point −t ·I

/θ ∈R

, and the angle of rotation −θ.

In parameterizing 3D conformal transformations, the 10D vector space Λ

4,1

)

provides an ideal parametric space for the group of 3D conformal transformations.

Since the exponential map from Λ

4,1

) to the group of positive rotors is surjec-

tive, any orientation-preserving 3D conformal transformation can be parameterized

via the exponential map, although the parameterization is not unique.

The problem of parameterizing with the exponential map lies in evaluating the

map and computing its inverse. While the map can be evaluated when restricted

to some vector subspaces such as Λ

∼

), the evaluation for the general case is

still not available. Even when the evaluation exists, in many cases such as (3), it is

computationally expensive because the map is transcendental instead of algebraic.

Furthermore, the exponential map is not an isometry, and its tangent map preserves

volume only at the origin of the Lie algebra taken as a vector space. The exponential

map has inﬁnitely many inverses in general.

The ﬁrst alternative of exponential map is the following exterior exponential map:

Deﬁnition 1 Let V

be a vector space over a ﬁeld K.Theexterior exponential is

the following map from Λ

) to Λ(V

∧B

=1 +B

∧B

+···+

  

∧B

∧···∧B

, (5)

where r is the greatest integer such that

  

∧B

∧···∧B

=0.

The exterior exponential has two obvious properties: ﬁrst, the scalar part of e

∧B

is 1; second, the mapping is injective because the bivector part of e

∧B

is B

Since any bivector has a completely orthogonal decomposition, i.e., for a bivector

, there exist vectors a

, b

, a

, b

,...,a

, b

such that

=λ

∧b

+λ

∧b

+···+λ

∧b

, (6)

where a

·b

= 0 for any 1 ≤ i, j ≤ r, and a

·a

= b

·b

= 0 for any i = k,we

have

∧B

=(1 +λ

∧b

)(1 +λ

∧b

) ···(1 +λ

∧b

So e

∧B

is invertible if and only if each λ

∧b

is not a Minkowski blade of unit

magnitude. When V

is Minkowski or Euclidean, then if e

∧B

is invertible, it must

76 H. Li

be a rotor connected with the identity, because so is each 1 +λ

∧b

; furthermore,



∧B



−1

∧(−B

)

∧B

∧(−B

)

. (7)

Below we assume that B

is in the form of (6) and e

∧B

is invertible, and analyze

the range of the exterior exponential by restricting it to the conformal model R

4,1

3D geometry.

If B

is a nonzero blade, then e

∧B

=1 +λ

∧b

. When λ

varies, the range

of e

∧B

modulo scale contains all rotors in Λ(a

∧ b

) whose 0-graded part and

2-graded part are both nonzero.

If B

is not a blade, then

∧B

=1 +λ

∧b

+λ

∧b

+λ

∧b

∧a

∧b

, (8)

whose 0-graded part and 4-graded part are both nonzero. The range of e

∧B

modulo

scale is all rotors whose 0-graded part and 4-graded part are both nonzero.

Proposition 1 When the range of the exterior exponential is restricted to rotors in

C(R

4,1

), the domain of deﬁnition is all bivectors in Λ(R

4,1

) satisfying

∧B

)

=4(B

·B

−1) (9)

and is a set R

−V

, where V

is a 9D algebraic variety in R

. The image space

modulo scale is all rotors whose scalar parts are nonzero; topologically, it is the

remainder of the positive orthogonal group O

(4, 1), which is a 10D Lie group

with two connected components, after removal of a 9D closed subset.

Proof By

∧B

=1 +B

∧B

, (10)

we get e

∧B

∧(−B

)

=1 −B

·B

+(B

∧B

)

/4, and (9) follows. 

Similar to the exponential map, the exterior exponential provides half-scaled

bivector representations for rotations, translations, and dilations.

Given a rotor A in the image space of the exterior exponential, let B

be a bivector

whose exterior exponential equals A up to scale. Then

1 +B

∧B

A

, (11)

A

. (12)

Parametrization of 3D Conformal Transformations 77

Example 3 Let e

be a rotor inducing the rotation in space with axis I

∼

and angle

−θ =π mod 2π. Then up to scale,

∧I

tan(

)

. (13)

Let e

=1 +et/2 be a rotor inducing the translation by a vector t. Then

∧

. (14)

Let e

be a rotor inducing the dilation of scale e

−θ

centering at point I

(afﬁne

representation, cf. [9]). Then up to scale,

∧I

tanh(

)

. (15)

Let I

be a rotor inducing a dilation of scale −e

−θ

=−1. Then up to scale,

∧I

ctanh(

)

. (16)

On one hand, the exterior exponential is an injective quadratic map, which is

superior to the exponential map algebraically. On the other hand, the exterior expo-

nential has two severe drawbacks: ﬁrst, the domain of deﬁnition is decomposed into

several disconnected regions, which blocks the construction of large-scope bivec-

tor parameters in the design of continuous conformal transformations; second, the

image space is also decomposed into several disconnected regions, making it im-

possible to represent rotors of large-scale continuous conformal transformations.

3 Twisted Vahlen Matrices and Quaternionic Vahlen Matrices

Recall that in complex analysis, any 2D conformal transformation can be repre-

sented by a fractional linear map from the Riemann sphere to itself, the sphere being

the complex plane plus the complex point at inﬁnity. In Clifford analysis, there is

a similar representation for any nD conformal transformation. This is the so-called

Vahlen matrix representation [1, 11, 12, 16].

In this section, we introduce the classical work of Vahlen (1902) on representing

nD conformal transformations projectively by 2 × 2 matrices whose components

are in C(R

), by means of introducing two alternatives of Vahlen’s matrix repre-

sentation:

• Twisted Vahlen matrices: the product of two 2 ×2 matrices of multivector com-

ponents is no longer the usual matrix product, but a “twisted” one. The twisted

matrix product is exactly the geometric product in the conformal geometric alge-

bra C(R

n+1,1

• Quaternionic Vahlen matrices: When n = 3, any twisted Vahlen matrix can be

written as a 2 ×2 quaternionic matrix, and projectively, any 3D conformal trans-

formation can be represented by such a quaternionic matrix. The composition of

3D conformal transformations is just the usual matrix product.